The Spaniard, a two-time winner at SW19, had hauled himself back from two sets down against the 16th seed from Luxembourg to take their last-16 encounter to a decider.
But Muller, who was the first player to beat a raw Nadal at Wimbledon back in 2005, repeated the trick 12 years later by triumphing 15-13 in an epic fifth set which lasted two and a quarter hours.
By contrast, Roger Federer continued his serene progress with a straight-sets win over Grigor Dimitrov.
The Swiss star, yet to drop a set, stayed on course for a record eighth All England Club title with a straightforward 6-4 6-2 6-4 victory in one hour and 37 minutes.
Dimitrov, the 13th seed from Bulgaria, was supposed to provide a tough test for the third seed on Centre Court.
Nicknamed 'Baby Fed' as a youngster for his resemblance in playing style to the 18-time major champion, Dimitrov was a semi-finalist at Queen's and has looked in good touch at SW19.
But Federer scarcely had to break into a sweat as he cruised into the quarter-finals.
Dimitrov had never beaten Federer in five previous attempts, but this was their first meeting on grass.
A tight first set went with serve until, in time-honoured fashion, Federer applied the pressure and secured his break in the ninth game before serving out the set.
Dimitrov, a semi-finalist in 2014, never really recovered as Federer simply moved through the gears.
Already a break up, a dropped service game in the third caused minor irritation for Federer, but he promptly broke back to snuff out any chance of a Dimitrov comeback.
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Milos Raonic, who beat Federer in last year's semi-final, awaits in the last eight this time around after he came from two sets to one down to beat Alexander Zverev.
Marin Cilic reached his fourth consecutive quarter-final with a straight-sets win over Roberto Bautista Agut.
The Croatian seventh seed overpowered his Spanish opponent in a one-sided contest, running out a 6-2 6-2 6-2 winner on Court 12.
Bautista Agut, seeded 18, had no answer to the fearsome groundstrokes of Cilic, one of the few men outside the top four to win a grand slam in recent years with his 2014 US Open victory.
The 28-year-old has never been beyond the last eight at these championships but looks every bit a threat to the big guns.
Sam Querrey came through the battle of the big servers against Kevin Anderson in five sets on Court 18 to set up a quarter-final against Andy Murray, who came through in straight sets against Benoit Paire.
The American lost the first set 7-5, but levelled on a tie-break before taking the third with a rare break of serve.
A marathon tie-break in the fourth went Anderson's way, after he survived four match points, but he was broken again in the decider as Querrey won 5-7 7-6 (7/5) 6-3 6-7 (11/13) 6-3.
Three-time champion Novak Djokovic. who had been due to follow the Nadal-Muller match on Court One, never even made it on court and must come back on Tuesday to play Adrian Mannarino of France.
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